What is an example of a contractible subset of $\mathbb{R}^N$ that is not a deformation retract of $\mathbb{R}^N$?
I don't know how to construct such a subset. Compact subsets of $\mathbb{R}^N$ and convex subsets of $\mathbb{R}^N$ always seem to be deformation retracts of $\mathbb{R}^N$. I also tried to start with $N = 1$ or $N = 2$, but I have not succeeded in finding such a subset.
Best Answer
The open unit ball is not a deformation retract since this would give a retraction of the closed ball into the open ball, but this is impossible since this is the identity on a dense subset of a Hausdorff space, and so is the identity everywhere.
Note this is stronger than showing it is not a deformation retract.